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Computer Science > Computational Complexity

Abstract: In this paper, we prove tight lower bounds on the smallest degree of a
nonzero polynomial in the ideal generated by $MOD_q$ or $\neg MOD_q$ in the
polynomial ring $F_p[x_1, \ldots, x_n]/(x_1^2 = x_1, \ldots, x_n^2 = x_n)$,
$p,q$ are coprime, which is called \emph{immunity} over $F_p$. The immunity of
$MOD_q$ is lower bounded by $\lfloor (n+1)/2 \rfloor$, which is achievable when
$n$ is a multiple of $2q$; the immunity of $\neg MOD_q$ is exactly $\lfloor
(n+q-1)/q \rfloor$ for every $q$ and $n$. Our result improves the previous
bound $\lfloor \frac{n}{2(q-1)} \rfloor$ by Green.
We observe how immunity over $F_p$ is related to $\acc$ circuit lower bound.
For example, if the immunity of $f$ over $F_p$ is lower bounded by $n/2 -
o(\sqrt{n})$, and $|1_f| = Ω(2^n)$, then $f$ requires $\acc$ circuit of
exponential size to compute.